# - Acharya T.

ISBN 0-471-48422-9

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In Chapter 4 we discussed the theoretical background of the DWT and its implementation using both a convolution approach as well as a lifting-based approach. The standard supports both the convolution and the lifting-based approach for DWT. We also discussed issues of VLSI implementations of the DWT for both convolution and lifting-based approaches in Chapter 5. For details of DWT and their VLSI implementations, the reader is referred to Chapters 4 and 5 respectively. In the rest of this section, we present the two default wavelet filter pairs supported by Part 1 of the JPEG2000 standard.

6.6.1.1 Discrete Wavelet Transformation for Lossy Compression For lossy compression, the default wavelet filter used in the JPEG2000 standard is the Daubechies (9, 7) biorthogonal spline filter. By (9, 7) we indicate that the analysis filter is formed by a 9-tap low-pass FIR filter and a 7-tap high-pass FIR filter. Both filters are symmetric. The analysis filter coefficients (for forward transformation) are as follows:

150 JPEG2000 S TA NDA RD

• 9-tap low-pass filter: [/i_4, /i_3, /i_2, /1-1, ^o, /11, h2, /13, ^4]

+0.026748757410810 -0.016864118442875

/14 = h—4 h 3 = /i_3

/i2 = h—2 h\ =

hç,

-0.078223266528988

+0.266864118442872

+0.602949018236358

• 7-tap high-pass filter: [g_3, g_2, 9-1, 90, 9i, 92, 93]

93 = g_ 3 = +0.0912717631142495

ff2 = ff_2 = -0.057543526228500

9l = g_Y = -0.591271763114247

go = +1.115087052456994

For the synthesis filter pair used for inverse transformation, the low-pass FIR filter has seven filter coefficients and the high-pass FIR filter has nine coefficients. The corresponding synthesis filter coefficients are as follows:

• 7-tap low-pass filter: [h'_3, h'_2, h'^, h'0, h[, h'2, h3]

h'3 = h’_3 = -0.0912717631142495

h'2 = hf_2 = -0.057543526228500

h[ = h’_i = +0.591271763114247

h’o = +1.115087052456994

9-tap high-pass filter: [g'_4, 9-3> 9-2, 9-i> 9o> 9i> 92i 93> 94]

94 = 9-4 = +0.026748757410810

93 = 9-3 = +0.016864118442875

92 = 9-2 = -0.078223266528988

9\ ~ 9-i = -0.266864118442872

9o = +0.602949018236358

For lifting implementation, the (9, 7) wavelet filter pair can be factorized into a sequence of primal and dual lifting as explained in Chapter 4. The detailed explanation on the principles of lifting factorization of the wavelet filters has been presented in Section 4.4.4 in Chapter 4. The most efficient factorization of the polyphase matrix for the (9, 7) filter is as follows [10]:

P(z) =

where a = -1.586134342, b = -0.05298011854, c = 0.8829110762, d = -0.4435068522, K= 1.149604398.

1 q(1 + z 1) ’ 1 0 ' ' 1 c(l + z""1) ' Ï 1 0 [ k o'

0 1 b(l + 2) 1 0 1 [ d(l + 2) 1 L 0 * .

COMPRESSION 151

6.6.1.2 Reversible Wavelet Transform for Lossless Compression For lossless compression, the default wavelet filter used in the JPEG2000 standard is the Le Gall (5, 3) spline filter [28]. Although this is the default filter for lossless transformation, it can be applied in lossy compression as well. However, experimentally it has been observed that the (9, 7) filter produces better visual quality and compression efficiency in lossy mode than the (5, 3) filter. The analysis filter coefficients for the (5, 3) filter are as follows:

• 5-tap low-pass filter: [/i_2, /i-i, ho, hi, h2\

h2 — h_ 2 = —1/8

h\ = /i_ i = 1/4 h0 = 3/4

• 3-tap high-pass filter: [g_i, g0, gi]

91 = 9-i = -1/2

5o = 1

The corresponding synthesis filter coefficients are as follows:

• 3-tap low-pass filter: [h'_i, h'Q, h\)

h[ = h'_i = 1/2 K = 1

• 5-tap high-pass filter: [g'_2, gLu g'0, g'i, g'2]

92 = 9-2 = “VS

9i=9-1 = -1/4

9o - 3/4

The effective lifting factorization of the polyphase matrix for the (5, 3) filter has been derived in Section 4.4.4 in Chapter 4. This is shown below for the sake of completeness:

‘1 £(1 + *)- 1 0 1

0 1 L -è(1 + ^_1) 1J

6.6.1.3 Boundary Handling Like a convolution, filtering is applied to the input samples by multiplying the filter coefficients with the input samples and accumulating the results. Since these filters are not causal, they cause discontinuities at the tile boundaries and create visible artifacts at the image boundaries as well. This introduces the dilemma of what to do at the boundaries. In order to reduce discontinuities in tile boundaries or reduce

152 JPEG2000 S TA NDA RD

artifacts at image boundaries, the input samples should be first extended periodically at both sides of the input boundaries before applying the onedimensional filtering both during row-wise and column-wise computation. By symmetrical/mirror extension of the data around the boundaries, one is able to deal with the noncausal nature of the filters and avoid edge effects. The number of additional samples needed to extend the boundaries of the input data is dependent on filter length. The general idea of period extension of the finite-length signal boundaries is explained by the following two examples.

Example 1: Consider the finite-length input signal ABCDEFGH. For an FIR filter of odd length, the signal can be extended periodically as

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